Category: Formulas

Many people are familiar with the different tax treatment between Traditional IRAs and Roth IRAs; unfortunately, often times most individuals are not sure which contribution type is right for them.

Typically, advocates for either type of IRA tend to elect 100% of their annual contribution limits towards their IRA flavor of choice.

Ultimately, if taxes never changed then mathematically it makes zero difference if you elected Traditional IRA or Roth IRA contributions, the ending balances would be the same.

Recall, that the future value of a present value today can be solved for with the following formula:

Where:
FV = Future Value
PV = Present Value
r = rate
n = period

In order to determine the after-tax value of a Traditional IRA or a Roth IRA, a new quantity for taxes is added. The key difference is the order that the taxes are taken out, which becomes clearer when comparing the two formulas. Here is the formula for the after-tax value of a Traditional IRA:

Where:
t = taxes

Comparatively, here is the formula for the after-tax value of a Roth IRA (which can also be referred to as the future value of a tax-free account):

Notice how the quantity (1 – t) is moved to the front of the equation, which makes sense given that Roth IRA contributions are taxed up front, and the remaining amount is invested. Since multiplication is cumulative, then the ending balances after paying taxes should be identical between both accounts.

Let’s assume that you invest $1,000 in a Traditional IRA growing at a rate of 7.00% for 10 years, when you retire your marginal tax bracket is 30%. Plugging those values into the formula for a Traditional IRA would yield the following:

Your initial $1,000 contribution grew tax free for 10 years, and after paying a 30% tax in the tenth year you are left with $1,377.01. The table below illustrates the tax free growth for 10 years, the tax paid, and the net after-tax future value:

Traditional IRA
Year	PV	Rate	FV	Tax (30%)
1	$  1,000.00	7.00%	$            1,070.00
2	$  1,070.00	7.00%	$            1,144.90
3	$  1,144.90	7.00%	$            1,225.04
4	$  1,225.04	7.00%	$            1,310.80
5	$  1,310.80	7.00%	$            1,402.55
6	$  1,402.55	7.00%	$            1,500.73
7	$  1,500.73	7.00%	$            1,605.78
8	$  1,605.78	7.00%	$            1,718.19
9	$  1,718.19	7.00%	$            1,838.46
10	$  1,838.46	7.00%	$            1,967.15
TAX				$  590.15
Net ATFV			$            1,377.01

We can represent the data above in the following chart:

Let’s plug the same values into the Roth IRA formula to prove that they are identical:

Despite starting with a lower initial investment since the tax with Roth IRA contributions are paid up front, the ending value is still identical to the Traditional IRA account:

Roth IRA
Year	PV	Rate	FV	Tax (30%)
1	$     700.00	7.00%	$     749.00
2	$     749.00	7.00%	$     801.43
3	$     801.43	7.00%	$     857.53
4	$     857.53	7.00%	$     917.56
5	$     917.56	7.00%	$     981.79
6	$     981.79	7.00%	$  1,050.51
7	$  1,050.51	7.00%	$  1,124.05
8	$  1,124.05	7.00%	$  1,202.73
9	$  1,202.73	7.00%	$  1,286.92
10	$  1,286.92	7.00%	$  1,377.01
TAX				$           –
Net ATFV			$  1,377.01

To reiterate, if your marginal tax bracket in the future is identical to your marginal tax bracket today, there is zero difference between a Traditional IRA and a Roth IRA (there may be a perceived difference due to emotional or cognitive biases).

Obviously, the future state of taxation is a huge assumption to make and the world is not that simple. There are a multitude of additional variables and factors that need to be taken into account in order to determine what blend between Traditional IRA or Roth IRA contributions makes the most financial sense for you.

Since there are exactly zero people on the planet who can accurately predict what the marginal tax brackets will look like when you retire (I’d be skeptical of anyone claiming that they can), having money in both account types will provide you with the maximum flexibility in determining your actual tax bracket when you retire.

Remember that the next time anyone says, “Roth IRAs are better than Traditional IRAs so I put 100% of my annual contribution limits into my Roth IRA”, or vice versa. Generalized advice, generally has good intentions, but produces generally bad outcomes, generally speaking.

Taxes aren’t the only variable that should be taken into account when determining the optimal Traditional IRA and Roth IRA blend. One should also consider the following:

• what will your burn rate be when you retire
• what will your projected federal tax bracket be
• do you pay state income taxes
• will you move to a state that has no income taxes
• will you move to a state that does have an income tax
• what is your current blend of qualified vs non-qualified assets
• what proportion of your employer matching contributions into any qualified plans is made on your behalf

The list above, while somewhat lengthy, is nowhere near exhaustive despite the potential to get exhausted while thinking about it.

All of the concepts above are also applicable to the world of 401(k)s. From a taxation standpoint, Traditional IRAs (which are a type of qualified accounts) are taxed identically to Traditional 401(k) contributions. Similarly, Roth IRAs are taxed identically to Roth 401(k) contributions.

Other important subtleties that you should be aware of for Roth IRAs and Roth 401(k)s are the lack of required minimum distributions. RMDs do not apply to Roth IRAs or Roth 401(k)s which make them an extremely potent estate planning wealth transfer tool.

In my opinion, the best kept secret about Roth 401(k) contributions are the fact that the income limitations that apply to Roth IRAs, do not apply to Roth 401(k) contributions. You could make $1mm dollars a year and still contribute the IRS maximum into a Roth 401(k).

If you don’t have a Traditional IRA or a Roth IRA you can open one at Charles Schwab for commission free trading with an initial funding bonus of up to $500 by clicking here.